Resolves as YES if there is strong evidence that OpenAI discovered the first valid proof of the Riemann Hypothesis before January 1st 2027. Only proofs made public before 2027 are considered, and the public release date determines which proof is deemed the “first.” If at least one candidate proof from OpenAI is made public before the end date but its validity remains uncertain, the end date can be extended until the proof’s validity is resolved. Both proofs that demonstrate the hypothesis or demonstrate that it's undecidable are acceptable in the context of this question. Proofs that the hypothesis is false are also acceptable.
If another entity makes a proof public first and that proof is considered valid, this question automatically resolves as NO. Similarly, if no publicly known OpenAI proof emerges before January 1st 2027, the question resolves as NO.
Examples of what qualifies as a "proof from OpenAI” include, but are not limited to:
Papers published to arXiv (or other preprint servers) with OpenAI researchers listed as authors.
Peer-reviewed journal or conference publications with (strongly) OpenAI-affiliated authors.
White papers or technical reports released on official OpenAI channels (e.g., blogs, GitHub, or websites).
Work using proprietary OpenAI models (not publicly available), where the researchers have privileged access and rely on OpenAI’s own compute resources or those provided through partnerships (e.g., Microsoft). Proofs discovered by running on third-party servers outside such partnerships (e.g., a university cluster) do not qualify under this condition.
A public release of any document or presentation that claims to prove the Riemann Hypothesis—produced (or co-produced) by OpenAI researchers and using OpenAI resources—will be sufficient to trigger a YES resolution if no other valid proof has been released prior to it. If the validity of an OpenAI proof remains uncertain by January 1st 2027, the resolution date may be postponed until its validity is established.
1,000People are also trading
@AleisterCrowley I'm not qualified whatsoever but legit-seeming people claim that for Y-M they have once again basically just scooped someone for the last mile:
Juspreet Singh Sandhu (@freegaussians.bsky.social)
https://nitter.xitter.cc/__alpoge__/status/2114841993847464207
(And this does not settle Y-M, since mass gap is still open and is the part where afaiu experts don't know where to start.)
@0xseraphim Like I said before, better to park mana in the market about all MPs. Was just waiting for enough NO to pile on here that I could exit without too much loss
@0xseraphim Yeah of course, since this is a subset of that. But that market is currently >33% and the difference between that and the true probability is much higher than here.
I freely admit I got wrong the question of whether a MP would be solved in 2026 "by AI". But I worry people outside of math are taking the wrong lesson away from it. At least half my error was to not be up to date on human developments in Navier-Stokes. (PDE was never my field.) Everyone in the field of PDE agrees that AI did something amazing but at the same time it was a "last few bricks in a castle" type thing.
I've researched the remaining 5 problems and my takeaway is there is no castle to add the last few bricks to. Nobody has any idea how to start. You know OpenAI tried all of them really hard for the 8 Sep release, and all they got was extremely narrow cases of BSD and Hodge; nothing towards Y-M or P vs NP; and a small advancement towards RH which was a combination of already-existing human ideas.
@0xseraphim If AI does it, I'm almost certain it will be by combining existing ideas, because in every example I've been able to verify for myself so far, and all examples which I heard about from professional mathematicians, that's what it was doing. I'm not saying this out of some disdain for AI; it's just what you should expect from knowing how current AI works.
So, I believe current AI can only solve one the remaining MPs if a solution exists which "just" combines existing ideas. Moreover, even if it does, there is the question of how long it is; and the fact that none appeared in the 6 Oct drop means that, even if such a solution does exist, it must be very long indeed.
@pietrokc how would you qualify the famous proofs of Fermat's last theorem and the Poincaré conjecture in that framing?
@AleisterCrowley I don't enough algebraic number theory to credibly comment on Wiles's proof of FLT. From reading popularizations, it seems to revolve around a connection between two kinds of objects ("elliptic curves" and "modular forms") which appear quite different at first. I find it easy to believe that the idea to relate these two things, whoever it is due to, was fundamentally new.
As for Poincaré, I believe the program proposed by Hamilton -- define a metric on the manifold that evolves via a heat diffusion type PDE and follow it to the end -- was based on a fundamentally new idea. Perelman was the one to finish this program, and I don't know enough about his proof to say whether other fundamentally new ideas were needed.
I don't know whether Hamilton was the first one to introduce this idea; chatbots suggest there were precursors in some form. But there is, I believe, an irreducible kernel of novelty somewhere, which I can explain with the following analogy. Imagine you spent years studying euclidean geometry. You become master of points, lines, and circles, you know all the theorems and proof techniques in The Elements. Well, the thing is that all the figures and ideas in Euclid are static. The circles and lines are fixed in place. Then one day someone solves a major open question in euclidean geometry by making the circles and lines move according to some equation. That's kind of (very roughly) what happened with Poincaré.
Of course, to us, the idea that geometric objects can move and evolve is very obvious. We get the very idea of geometry from the real world, and stuff in the real world moves constantly. But if, in a supreme exercise of empathy and "putting yourself in someone else's shoes", you imagine you are a machine, and you have no notion of the real world, or space, or time, and you are simply RLed on a billion facts about static lines and circles -- I don't think you would ever conceive of moving them around.
@pietrokc then how do you interpret move 37 (from alphago)? Where did it come from? What is your thinking on what something similar would look like in mathematics?
@pietrokc because alphago didn't know anything about physics or military strategy or whatever else human players derive their intuition from. And yet it figured out unique styles of play that people then spent years studying and imitating
@AleisterCrowley These are great questions!
One important bit of context, which I've mentioned on this site before but maybe not to you, is this. It has been known since the 1930s that it is possible to write a (fairly short) computer program which will (1) sequentially output all theorems provable from the standard axioms of mathematics (called ZFC); and (2) never output any statements NOT provable from them.
So, although it is often forgotten in online debate, we have known for 100 years that a computer could in theory solve all millennium problems and much more. The only issue is how long it would take. The program in question is extremely slow, and would take many lifetimes of the universe to prove even simple theorems from undergrad. Because the way that it works is not by "understanding" any math, but simply by running over all possible strings of 0s and 1s and checking, for each one, whether it encodes a proof of a theorem. If a statement is provable, then it has a proof, that proof can be encoded as a sequence of 0s and 1s, and the computer program will get to it eventually.
In other words, we have known for 100 years that computers can create "new math". The issue is without some hint of which direction to go in, all they can do is create ALL math, of which some will be new.
You may ask, well, how can humans know the direction to go in? And my reply is because math comes from our own brains. In a way we're grading ourselves on a test that we ourselves devised. That's why we have an advantage. (There may be other reasons.)
To tie this to your question about Go: AlphaGo was trained in a way reminiscent of the all-proving computer. It played itself billions of times, and in some sense it "tried everything", like the proving program just tries all sequences of 0s and 1s, and that's how it found something new. (In reality it didn't try everything, only A LOT more things than any human ever has. But note that no one claims move 37 was best possible! That is almost certainly not true, and unknowable. The only claim is that it was superhuman at the time.)
Importantly, a computer was able to "try everything" in Go because in Go there is nothing beyond the stones and their placement. So it knew the space of possible things to try. Whereas with math we may write down the definition of lines and circles, but we have a lot more in our heads that goes unwritten, because as humans we all share the same "brain shape" and we all "get it" -- for instance, that lines and circles could move. The analogy is more like if the computer was trained on Go, then one day, while playing against a human, the human declared that since stones are small and round, they can be picked up and eaten. Or if you took took the Go computer and assumed that, since it "understands" concepts like outnumbering and positioning, it would be good at Starcraft.
Incidentally, idk anything about Go, but chatbots tell me physics and military strategy are not helpful for Go mastery; that most high level players start learning between ages 5-9 and become masters long before they had a chance to study either physics or military strategy. This supports the view that Go is self-contained, which lets computers "try everything" in a meaningful way.
@pietrokc I think this quote is the core of where I disagree:
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In other words, we have known for 100 years that computers can create "new math". The issue is without some hint of which direction to go in, all they can do is create ALL math, of which some will be new.
You may ask, well, how can humans know the direction to go in? And my reply is because math comes from our own brains. In a way we're grading ourselves on a test that we ourselves devised. That's why we have an advantage. (There may be other reasons.)
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@pietrokc Machines are no less capable than human brains at judging direction. There's nothing special about biological computers.
@AleisterCrowley That claim is a popular soundbite but it is incorrect. Whether it's true depends on the specific task.
For instance, electronic computers are evidently much better at multiplying large numbers. It would be silly to claim that "there is nothing special about electronic computers".
On the other hand, biological computers are evidently much better at, say, simulating biological computers. (In fact, something like this was the original motivation for quantum computers! That they would be much better at simulating quantum phenomena.)
My whole point is that mathematics is not some random thing in the universe for which there is no reason to expect electronic or biological computers to be better. Mathematics is a product of the biological computer; it can even be argued to be a description of how the biological computer operates. (You may research Kant's philosophy of mathematics as a start on this topic.)
As a loose analogy, it would be silly to claim "there's nothing special about biological computers" when it comes to the question of whether you like chocolate. You can answer that question in half a second with your brain that runs on about 20 watts, in some sense because it is a question about the brain itself.
@AleisterCrowley As a separate direction of counterargument, the "nothing special" framing also does not distinguish between types of electronic computer. If there is "nothing special" about biological computers re math, why did it take until 2026 for electronic computers to start catching up? Electronic computers have been MUCH faster (in MHz) than biological ones for 30+ years. There have been automated theorem provers since the 90s or even earlier.
@pietrokc You're getting confused and you're mixing up multiple different concepts. The main ones I can read from your comments are "where does maths come from?", "what makes a system good at discovering maths?", "can such a system simulate a brain?" and "how energy efficient is this system?". These are distinct questions and you're sliding between them as if they were all the same thing. Just because a system is good at one of these things doesn't make it good at the others.
So first of all, just because you invent at task, it doesn't mean that you're the best at it. For example, humans invented multiplication (which is maths), and you point out that computers outperform humans on it. So your argument that "mathematics is a product of biological brains" is not enough to explain a biological advantage regarding maths discovery. You're missing another argument about what makes discovery in maths different, and why the key abilities there require biology. The maths we've got is for sure a byproduct of human thought, and it's shaped by what interests us and our intuitions. So from an "initial starting condition" in 2026 humans have (to an extent) got a useful starting bias. But this bias in no way translates into an unbeatable advantage. And you've failed to demonstrate in your argument that machines can't learn those biases or develop better ones.
@pietrokc Then the second thing is that your "why did computers take so long?" argument is a double-edged sword. Why do you think humans pissed about for 100k+ years before coming up with the fourier transform? Why was didn't the first humans with modern homo-sapiens brain anatomy figure it out? You can't have it both ways. If the super long delay in humans making progress in maths doesn't establish a core limitation of biological hardware, why shoud a dramatically shorter delay in computer progress establish a core limitation of electronic hardware? Your argument doesn't hold up. Processors need algorithms/representations/training methods etc that build up and improve over time in order to get good at maths. Same way humans need accumulated knowledge and conceptual frameworks they develop over thousands and thousandsof years
Next issue is that you're confusing processor clock speed with effective computation. Processor frequency and neuron firing rates aren't great measures of mathematical capability 🤣 . Just because a fast electronic computer running proof-search software fails on a given objective doesn't mean that electronic hardware is inherently unsuited to mathematics. That argument doesn't hold up at all
@pietrokc The chocolate analogy (by my reading of it) is a mix up between two very different concepts: "accessing an internal state" and "solving a maths problem". You're implicitly talking about two distinct things as if they're the same thing and that's tripping you up. Preferences are states of the brain, which is why the brain has access to those preferences. What you're saying in your analogy is you're basically trying to claim that proofs in maths are some sort of introspection in a similar fashion to how we access these preferences. That doesn't make sense at all. And then even if you squint really hard and decide that mathematical cognition is some "Kantian process", that still doesn't let you claim that electronic systems couldn't implement the same representations and reasoning. Like just because you hypothetically solve "where do concepts in maths come from?" that doesn't mean you suddenly get the answer to "which hardware can better reason about these concepts?"
Also machines aren't solely limited to exchaustive search. The description you gave before of alphago missses the relevant point. It has two parts: a policy network that learns which moves to take into account and a value network that learns which configurations are important. If you read the original paper then you'll see that they even report effective playing ability even without running the lookahead search. That is a machine learning how to judge which directions it should point its efforts towards (and not just scanning different possible outcomes with a higher clock frequency). This undermines the distinction you're trying to establish between humans who have a "sense of direction" and computers that are just basic brute force machines
@pietrokc Then finally for the brain simulation and energy efficiency stuff, you're essentially conflating the concept of "being a biological system" and "simulating a biological system" while trying to smoosh together "energy efficiency' and "present + future capability". Basically your analogy to quantum computing just doesn't work. Brains don't simulate other brains in detail; Brains embody a physical process that sometimes produces mathematical proofs. In order to discover a mathematical proof you don't need to reproduce the exact biology of a mathematician's brain. Electronic computers can already discover these proofs without needing this embodiment. The analogy is lacking some mechanism is human brains that is key for mathematical discovery you can point to that is prohibitively expensive to reproduce on a digital computer. You just stated the analogy without defining what that mechanism was, so there's no argument here. And for sure brains are currently more energy efficient than electronic processors. But that doesn't mean machines can't produce better mathematics. Humans might have good intuitions in some areas making them still relevant actors in maths, but that in no way supports the claim that they're going to have some "permanent advantage". It just doesn't hold up
Think of it this way: if you can't find a mechanism to point to in order to support this analogy, it's probably because that mechanism doesn't exist